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Study on representations of algebraic groups and finite groups, and applications

代数群和有限群的表示研究及应用

基本信息

批准号：
12640040
负责人：
SHINODA Ken-ichi
金额：
$ 2.5万
依托单位：
Sophia University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640040/
关键词：
character sum Gauss sum Kloosterman sum Hilbert scheme of G-orbits coinvariant algebra McKay correspondence algebraic group finite group gaussian sum unitary Kloosterman sum Gelfand-Graev representation McKay correspndence Hilbert

项目摘要

1. We studied the coinvariant algebras of finite subgroups G of SL(3, C). Particularly for the simple groups G of order 60, 168, we investigated the fiber of Hilb^<|G|>(C^3) over the origin (joint work with Nakamura (Hokkaido U.)). Also we showed that for the subgroups of SL(3, R), the fibers are the finite union of P^1. As a consequence of these studies, we could also show that the relations of Molien series and representation graphs, which have been studied by Springer, McKay, etc, for finite subgroups of SL(2, C), can be generalized for finite subgroups of SL(n, C) using the Koszul complex. (Shinoda, Gomi)2. Shinoda studied the properties of Gauss sums for finite reductive groups applying the theory of Deligne-Lustzig. Particularly for the semisimple representaions of classical groups and for the unipotent representations of G_2(q), we had explicit formula for the corresponding Gauss sums (jointly with N. Saito (Sophia U.)).3. We also obtained the following results :(1) Tsuzuki studied the Shintani functions on real Lie group U(n, 1) obtaining an explicit formula of the Shintani functions and also some multiplicity-free theorem for the corresponding represntation.(2) Nakashima realized irreducible representaions of finite dimensional quantum algebras of type A at root of unity through the representations of quantum group U_ε of type A of non-restricted specialization.(3) Goto clarified the structure of Goodman-de la Harpe-Jones subfactors using combinatorial methods.(4) Koga realized Wakimoto representation for the affine Lie superalgebra of type A and obtained a charcter formula for the heighest weight, representations (jointly with K. Iohara (Kobe U.)).

1. 研究了SL（3， C）的有限子群G的协不变代数。特别是对于60,168阶的简单群G，我们研究了Hilb^<|G|>(C^3)在原点上的纤维（与Nakamura （Hokkaido U.）合作）。我们还证明了对于SL（3， R）的子群，纤维是P^1的有限并。作为这些研究的结果，我们也证明了施普林格、McKay等研究的SL（2， C）有限子群的Molien级数和表示图的关系，可以用Koszul复形推广到SL（n， C）的有限子群。(信田,无味)2。Shinoda运用delign - lustzig理论研究了有限约化群高斯和的性质。特别是对于经典群的半简单表示和G_2(q)的幂偶表示，我们（与N. Saito （Sophia U.）共同）给出了相应的高斯和的显式公式。(1) Tsuzuki研究了实李群U（n, 1）上的Shintani函数，得到了Shintani函数的显式公式，并给出了相应表示的若干无重性定理。(2) Nakashima通过非限制专门化的A型量子群U_ε的表示实现了A型有限维量子代数在单位根上的不可约表示。(3) Goto用组合方法阐明了Goodman-de - la Harpe-Jones子因子的结构。(4) Koga实现了A型仿射李超代数的Wakimoto表示，并（与K. Iohara （Kobe U.）联合）得到了最高权值表示的特征公式。